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Properties

Label 24.0.89730717781...1129.1
Degree $24$
Signature $[0, 12]$
Discriminant $7^{20}\cdot 13^{18}$
Root discriminant $34.65$
Ramified primes $7, 13$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 972, 810, 189, -855, -1860, -2243, -1621, 239, 2767, 4818, 5040, 2506, -2547, 748, 180, -245, 153, 58, -3, -12, 7, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - x^22 + 7*x^21 - 12*x^20 - 3*x^19 + 58*x^18 + 153*x^17 - 245*x^16 + 180*x^15 + 748*x^14 - 2547*x^13 + 2506*x^12 + 5040*x^11 + 4818*x^10 + 2767*x^9 + 239*x^8 - 1621*x^7 - 2243*x^6 - 1860*x^5 - 855*x^4 + 189*x^3 + 810*x^2 + 972*x + 729)
 
gp: K = bnfinit(x^24 - x^23 - x^22 + 7*x^21 - 12*x^20 - 3*x^19 + 58*x^18 + 153*x^17 - 245*x^16 + 180*x^15 + 748*x^14 - 2547*x^13 + 2506*x^12 + 5040*x^11 + 4818*x^10 + 2767*x^9 + 239*x^8 - 1621*x^7 - 2243*x^6 - 1860*x^5 - 855*x^4 + 189*x^3 + 810*x^2 + 972*x + 729, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - x^{22} + 7 x^{21} - 12 x^{20} - 3 x^{19} + 58 x^{18} + 153 x^{17} - 245 x^{16} + 180 x^{15} + 748 x^{14} - 2547 x^{13} + 2506 x^{12} + 5040 x^{11} + 4818 x^{10} + 2767 x^{9} + 239 x^{8} - 1621 x^{7} - 2243 x^{6} - 1860 x^{5} - 855 x^{4} + 189 x^{3} + 810 x^{2} + 972 x + 729 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8973071778116912219457090949865341129=7^{20}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.65$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(66,·)$, $\chi_{91}(5,·)$, $\chi_{91}(8,·)$, $\chi_{91}(73,·)$, $\chi_{91}(12,·)$, $\chi_{91}(79,·)$, $\chi_{91}(18,·)$, $\chi_{91}(83,·)$, $\chi_{91}(86,·)$, $\chi_{91}(25,·)$, $\chi_{91}(90,·)$, $\chi_{91}(27,·)$, $\chi_{91}(31,·)$, $\chi_{91}(34,·)$, $\chi_{91}(38,·)$, $\chi_{91}(40,·)$, $\chi_{91}(44,·)$, $\chi_{91}(47,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(57,·)$, $\chi_{91}(60,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{4}$, $\frac{1}{1663355697} a^{19} - \frac{224718197}{1663355697} a^{18} - \frac{242292505}{1663355697} a^{17} + \frac{137277004}{1663355697} a^{16} + \frac{2887114}{1663355697} a^{15} - \frac{18004652}{1663355697} a^{14} + \frac{59770719}{554451899} a^{13} + \frac{44828101}{1663355697} a^{12} - \frac{155295897}{554451899} a^{11} - \frac{346642136}{1663355697} a^{10} - \frac{247854082}{554451899} a^{9} - \frac{561188549}{1663355697} a^{8} + \frac{132017291}{554451899} a^{7} + \frac{455358346}{1663355697} a^{6} - \frac{365702323}{1663355697} a^{5} + \frac{159072493}{554451899} a^{4} - \frac{305139659}{1663355697} a^{3} + \frac{134327888}{554451899} a^{2} - \frac{667890754}{1663355697} a - \frac{6009384}{554451899}$, $\frac{1}{4990067091} a^{20} - \frac{1}{4990067091} a^{19} + \frac{511919366}{4990067091} a^{18} + \frac{167082778}{4990067091} a^{17} + \frac{157478060}{1663355697} a^{16} - \frac{46919774}{554451899} a^{15} + \frac{273338317}{4990067091} a^{14} - \frac{74713328}{1663355697} a^{13} + \frac{1561116037}{4990067091} a^{12} - \frac{221719193}{1663355697} a^{11} - \frac{924379535}{4990067091} a^{10} - \frac{269518463}{1663355697} a^{9} + \frac{303403123}{4990067091} a^{8} - \frac{129445916}{1663355697} a^{7} + \frac{24030642}{554451899} a^{6} - \frac{790388714}{4990067091} a^{5} + \frac{2452963667}{4990067091} a^{4} - \frac{776742364}{4990067091} a^{3} + \frac{377211325}{4990067091} a^{2} + \frac{17482879}{1663355697} a + \frac{226012648}{554451899}$, $\frac{1}{14970201273} a^{21} - \frac{1}{14970201273} a^{20} - \frac{1}{14970201273} a^{19} + \frac{2155324624}{14970201273} a^{18} - \frac{263800522}{4990067091} a^{17} + \frac{490273136}{4990067091} a^{16} - \frac{1246851959}{14970201273} a^{15} - \frac{7532126}{554451899} a^{14} + \frac{974419468}{14970201273} a^{13} + \frac{466054}{554451899} a^{12} + \frac{1042481062}{14970201273} a^{11} + \frac{244730774}{554451899} a^{10} + \frac{3404583679}{14970201273} a^{9} + \frac{91938136}{554451899} a^{8} - \frac{20796890}{4990067091} a^{7} + \frac{3863074621}{14970201273} a^{6} - \frac{4022725630}{14970201273} a^{5} + \frac{2691572264}{14970201273} a^{4} + \frac{6931540672}{14970201273} a^{3} - \frac{24758285}{4990067091} a^{2} + \frac{450621374}{1663355697} a - \frac{157502619}{554451899}$, $\frac{1}{44910603819} a^{22} - \frac{1}{44910603819} a^{21} - \frac{1}{44910603819} a^{20} + \frac{7}{44910603819} a^{19} + \frac{42348166}{516213837} a^{18} - \frac{2388452266}{14970201273} a^{17} - \frac{203224055}{44910603819} a^{16} + \frac{607069928}{4990067091} a^{15} + \frac{139124059}{44910603819} a^{14} - \frac{594823318}{4990067091} a^{13} - \frac{2401051229}{44910603819} a^{12} - \frac{975954595}{4990067091} a^{11} - \frac{601021382}{44910603819} a^{10} - \frac{2254041673}{4990067091} a^{9} + \frac{3141078196}{14970201273} a^{8} + \frac{10185600244}{44910603819} a^{7} - \frac{18553297864}{44910603819} a^{6} + \frac{6912953261}{44910603819} a^{5} + \frac{13717186783}{44910603819} a^{4} + \frac{4823519194}{14970201273} a^{3} - \frac{389314259}{4990067091} a^{2} - \frac{262771699}{554451899} a + \frac{1385561}{554451899}$, $\frac{1}{134731811457} a^{23} - \frac{1}{134731811457} a^{22} - \frac{1}{134731811457} a^{21} + \frac{7}{134731811457} a^{20} - \frac{4}{44910603819} a^{19} + \frac{4895720243}{44910603819} a^{18} + \frac{1826590072}{134731811457} a^{17} + \frac{541380383}{14970201273} a^{16} + \frac{5312435542}{134731811457} a^{15} - \frac{766866724}{14970201273} a^{14} + \frac{10286853004}{134731811457} a^{13} + \frac{6117758186}{14970201273} a^{12} - \frac{11197894982}{134731811457} a^{11} - \frac{2066527060}{14970201273} a^{10} - \frac{7741441742}{44910603819} a^{9} - \frac{14817427187}{134731811457} a^{8} + \frac{20464788425}{134731811457} a^{7} - \frac{27931004335}{134731811457} a^{6} - \frac{2595305585}{134731811457} a^{5} + \frac{307786933}{44910603819} a^{4} + \frac{2892100477}{14970201273} a^{3} + \frac{21809458}{4990067091} a^{2} + \frac{15404526}{554451899} a + \frac{211991442}{554451899}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{40072496}{134731811457} a^{23} + \frac{70126868}{134731811457} a^{22} + \frac{10018124}{134731811457} a^{21} - \frac{310561844}{134731811457} a^{20} + \frac{234429025}{44910603819} a^{19} - \frac{80144992}{44910603819} a^{18} - \frac{2414367884}{134731811457} a^{17} - \frac{1462646104}{44910603819} a^{16} + \frac{14416080436}{134731811457} a^{15} - \frac{4858790140}{44910603819} a^{14} - \frac{24564440048}{134731811457} a^{13} + \frac{42601809713}{44910603819} a^{12} - \frac{176970160460}{134731811457} a^{11} - \frac{42216374536}{44910603819} a^{10} - \frac{13865083616}{44910603819} a^{9} + \frac{33921367864}{134731811457} a^{8} + \frac{73583120780}{134731811457} a^{7} + \frac{72140510924}{134731811457} a^{6} + \frac{365195693533}{134731811457} a^{5} + \frac{791431796}{14970201273} a^{4} - \frac{801449920}{4990067091} a^{3} - \frac{410743084}{1663355697} a^{2} - \frac{110199364}{554451899} a - \frac{60108744}{554451899} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 209675214.65936095 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-91}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-7}, \sqrt{13})\), 4.4.107653.1, 4.0.2197.1, 6.0.36924979.1, 6.6.5274997.1, \(\Q(\zeta_{7})\), 8.0.11589168409.1, 12.0.1363454074150441.1, 12.12.2995508600908518877.1, 12.0.61132828589969773.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$